The sad fact is, I was a star geometry student in high school, and honestly remember *very* few of the theorems we worked on: probably fewer than 10%. (It sharpened my deductive logic skills, though!) In my later years as a computer scientist, I became interested in computational geometry and analytic geometry, ie, geometry with numbers included!
I'm a firm believer in a visual approach to mathematics, particularly in graphing functions, etc. Geometry seems like a natural lead-in to visualizing math. For example, you can demonstrate and prove the pythagoream theorem by taking 4 similar right triangles and arranging them in a square with a hole inthe middle. Or, you can show that a^2 - b^2 = (a-b)(a+b), or that (a-b)^2= a^2-2*a*b+b^2, etc. Using geometric shapes to prove these theorems (as I suspect the Greeks et al. did) is just another rich avenue for learning. In particular, it grounds the abstract algebraic manipulations in something manipulative and visual.
I wouldn't want to eliminate geometry entirely, but I might change the mixture to make it more relevant to both college-bound and non-college-bound students. For instance, take a city map that's a grid of N-S and E-W streets. You're in a car on the NW corner, and want to drive to some location on the East side. You can show that ANY path you take, as long as you drive either South or East, toward your destination is the same length! This is a little counter-intuitive to students who memorized "the shortest distance between two points is a straight line". Car's can't drive in straight lines through a city, though.
Sorry for the long-windedness. This is the first time I've tried to put these thoughts on paper. Overall, I'd say we should study geometry (and not just the compass-straight-edge variety!) because:
1. It's an intellectually honest, important component of the development of mathematics. 2. One can prove a variety of algebraic, trig, and other properties using geometric constructions 3. It's one of many possible vehicles for demonstrating deductive logic. 4. Many "real world" problems can be modeled as geometric problems.
My criticisms of current geometry practices are:
1. It's not the *only* method for demonstrating proof. In fact, we might mis-lead students into thinking that only in geometry to we prove theorems; the rest of mathematics was created by black magic.
2. The compass-straight-edge model is an interesting intellectual exercise, but is *not* the sole essence of geometry. In particular, we should be combining algebra and geometry at an earlier stage.
As far as the "what use is it" questions, some of it's useful for modelling problems, and otherwise it's an intellectual strengthener. What "use" is an aerobics class? I mean, how often in the real world are you going to swing your arms and legs in that manner? Or bench-pressing? How often do you really have to lift a load of iron off your chest? See what I mean? I'm cautious about taking a too literal view of "what use is this" when it comes to mathematics. On the other hand, we do aerobics and lifting so that we can enjoy the other activities in our life more fully. Same should be true (intellectually) with mathematics.
Good luck with your discussion! I, too, would be interested in a synthesized summary of these responses.
Thank you for asking a provocative question,
Larry Gallagher Institute for Research on Learning firstname.lastname@example.org
This whole geometry debate has centered on the content of a Geometry course -- Euclidean vs. non-Euclidean vs 3-dimensional, etc. etc. I would like to point out an advantage that geometry offers in terms of the PROCESS of learning.
One pleasant effect of our geometry curriculum here is that it provides an excellent environment to introduce our students to the principles of inductive reasoning and cooperative learning and problem-solving. Because much of geometry can be done intuitively, it gives students a chance to focus a bit more closely on the social skills of classroom interaction. By the time they get to Precalculus and Calculus (when, one could argue, they NEED to be able to work together and reason inductively), their cooperative learning skills are honed to the point that they work together smoothly (and often regardless of whether the teachers asks them to or not!).
See Michael Serra's "Discovering Geometry: an Inductive approach" from Key Curriculum Press for more information.
Subject: Re: Why teach Geometry in HS?
In addition to geometry being useful in certain `real world' situations. I think that the subject, if taught correctly, can be used to develop creative and critical thinking skills. Analyzing and synthesizing information, and making conjectures about geometric principals will force students to think -which is something they don't do enough of as they coast through classes imitating and regurgitating (sp?). Geometry students should be active learners, and much of it can be done cooperatively as well.
The Geometer's Sketchpad is a great piece of software to help in these aims.
On Wed, 30 Mar 1994, Doug Hale wrote: > Regarding college level algebra/geometry -- hear! hear! > Doug Dear Doug et al, I have many notions that I would like to discuss on this issue. This point of why not a college level algebra/geometry is a rather touchy one. We are most fortunate to have a real *geometer* at NMSU; only ONE! My initial educational experience was in aeronautical engineering followed by YEEKS *PHILOSOPHY*!! Please note that it was the philosophy that brought me home to mathematics. Yes that's right! Maybe as math type people we can see the connection, but the formal world of education has lost the vision of what philosophy began as... Logical rhetoric is more than linguistics, Greek, and political interpretations in the marketplace. It is indeed the foundation of mathematical thought. When I interviewed departments at the University to conduct my Master's Degree, the department of Education laughed at my background. Their response "hmmm... Philosophy is not a teaching field!" Whatever happened to the historical background of education. Did they forget that PhD translates to Doctorate of Philosophy?? Maybe, they just were never told what they were earning. Anyway... The department of Mathematics did not laugh. They said "Wow, welcome aboard. Have you studied the tenets of formal logic?" I said, "Of course, Why???" They said, "Well Kathrine, you should be able to master Boolean Algebra rather well!" I said, "What's that? But I should tell you that I really understand Geometry which I have studied as a logical set that defines the characteristics of space." They said, "See Dr. Zund, He's our geometer!" So, I did. And was I ever glad. You have not taken geometry until you have taken a course from a real geometer. His specialty by the way, is in topology. But most of all, he was a teacher! We even went through high school texts and found geometric falacies in the proofs. I entered the world of secondary math education as a total sceptic. That scepticism has been reinforce over and over. We talk about the math anxiety of Elementary Educators, but finding someone who really wants to teach high school geometry is like pulling teeth. Now, for the rest of the story...
*We must love geometry and appreciate its logical and physical form in order to teach it to kids. Otherwise we will continue to reinforce its difficulty rather than its elegance of form, and thelogical thinking that it promotes.
*We must recognize that the roots of Algebra and its *raison d'etre* is to analyze geometric relationships. (Of course this is also true of Calculus which is merely the extension of Algebraic Analysis of more advanced space.)
*We must introduce our students to the difference in mathematical disciplines, which I delienate as either Numerical or Geometric. Often as mathematics people we mistakenly discuss this issue as the debate between Pure and Applied Mathematics. This is a false discussion, because there are both Pure and Applied mathematics in both the Numerical and Geometric disciplines.
*In accordance with the above, we must recognize the diversity of the Geometric discipline and its many spinoffs; e.g. fractals, manifolds, non-Euclidian space, topology (and its little sister, Knots) ...
I could go on and on! Next year, I will be using the Creative Publications Algebra Themes Tools and Concepts Curriculum. It proposes that the learning of Algebra should take place in a lab environment where manipulatives model the equations that analyze spatial characteristics and relationships. If our Elementary teachers are going to introduce our young mathematicians to the world of space and number through physical forms, then we must not drop the bucket and let the kids believe that the learning they gained through their hands on experience is not sophisticated enough, or the right stuff for really smart kids. I guess I better qualify all of this harangue with the fact that I am a *Visual Learner* and was always the student asking, show me a picture, and I don't see what that means! Not given the whole picture, I did not understand the connection between Algebra and Calculus and Geometry. If we better understand the set of rules that we are using to define our space, we are better able to analyze its complexity. Sorry folks... I told you this pushed some buttons!!! :-) Kathrine Graham in New Mexico email@example.com
As a Math/Science teacher at the H.S. level, this discussion must surely be equivalent to heated debate I imagine taking place over the value of formal logic courses , or the benefits of Latin as a language option in the modern High School.. circa 1935 perhaps ? Lets get real, I found my 2 years of latin probably the most valuable course of my high school career, for it support of English through understanding of cognates, and particularly in science, where upon jargon came to have meaning: I could make out the meaning of science terms fairly accurately just from recognizing their latin roots. Dozens of other
real benefits could be desribed, depending on a persons interest; languages, history, philosophy, religion-mythology,and so on. If one thinks of subjects naturally integrative of diverse disciplines Latin studies looks even brighter as a course offering. NOT ! The value of Geometry as a course of study would seem by analogy tohave little to do with its inclusion or non-inclusion in a curriculum ! I for one am tired of looking at H.S. curriculum as certain doses of Algebra, Biology, a litle shot of Physics, some hours of instruction in Geometry, Chemistry, Calculus, lets round things out with a course in Art, or Music. We do want well rounded intellects. If we can find a way to introduce the geometry in and inter-disciplinary context; fine, I'll lend my support. Otherwise we had better do a much better job of selling our courses than the Latin teachers. Smile to all :) Landon
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Reply to: RE>Why teach Geometry in HS pro and con Why we shouldn't teach geo , the proof course, in h.s.:
1. if we want to teach axiomatic reasoning, we'd be better off doing it with algebra or something with a smaller more well defined set of axioms. (after all, we did get non-euclidean geometry when folks started poking at some of euclid's givens)
2. All that time could be better spent exploring more informal verison of geometry that really do build on and extend our spatial reasoning.
3. Algebra and geometry should go hand and hand, not separated. That would make for better prep for calculus stuff too, for those who want that
Why we should teach geometry, the proof course, in h.s.: There is no better defense of it as a h.s. course than Chip Healy's recent book. It's working title was "build a book geometry", but it's got another title now. I think it's published by Dale Seymour. Healy, a teacher in a working class suburb of LA, has kids write their own geometry book. He starts them with 3 axioms, and some tools like Geomters Skecthpad. They determine the rules of argumentation, standards of proof and collect all the theorems that get proven over the course of a year. He writes not only about what the kdis do in class, but the remarkable effects this experience has on their lives outside of school.
Subject: Re: Why teach Geometry in HS?
Before responding to your email, Patrick, a couple of notes: - I am responding to only you, John Meseke, and you, Patrick Williams, because you started the debate & wrote the email, respectively. Is that right? Or should I be responding to everyone on the list? You asked, John, that we send our messages to you; are you planning to send this out to everyone eventually? I don't want to inconvenience you by forcing you to do my forwarding for me, but I don't want to undermine your position as moderator by sending everything I write to everybody. How exactly does this work? - Also I want you both to know that this response is in the spirit of the "devil's advocate" responses you asked for, John. Please don't take any of it personally, Patrick.
In defending geometry in high school, Patrick, you wrote of a few experiences in which geometry helped you (or would've helped your neighbor) in real life. It seems to me, though, that those experiences were not related to the geometry I see taught in high school. For instance, you talked about your dad's wrangling with the city over the price of his land, but the math it would take to do that wrangling is taught well before high school (e.g. finding the area of a parcel of land, or multiplying the $/acre to find its total value). Any elementary school grad should have those skills without high school geometry. And the other examples you gave: a neighbor losing his foundation, building a tree-fort, etc. are examples of engineering, not geometry. Granted, a good deal of geometry is used in engineering, but I don't think most geometry classes focus on the skills needed to determine if a foundation will be undermined by local excavation. Are you saying that we should learn geometry in case we choose to become engineers, or that we should forsake geometry for engineering classes in high school which would be more useful? And speaking of usefulness, how does construction fit with this (by construction I mean the staple of high school geometry: compass and straight-edge construction; perpendicular bisectors, circumscribed circles and the like)? Will something like that ever be useful outside of the final exam? Or proofs? Personally, I don't think many people use the geometry we teach in high school in the outside world. -Dug Steen